Table 8.

Identifying assumptions for ${\tau }_3=\mathrm{E}\left[Y_{1M_{\mid C,0,L_1}}\right]-\mathrm{E}\left[Y_0\right]$

	Assumptions	Related to
		$\mathrm{E}\left[Y_{1M_{\mid C,0,L_1}}\right]$	$\mathrm{E}\left[Y_0\right]$
Consistency
Potential outcomes	$Y=Y_{1m}$ if $A=1,M=m$	$✓$
	$Y=Y_0$ if $A=0$		$✓$
Potential mediator	$M=M_0$ if $A=0$	$✓$
Potential int. confnders	$L=A{L}_1+\left(1-A\right)L_0$	$✓$
Conditional independence
Exposure-mediator	$A\perp \perp M_0\mid C$	$✓$
Exposure-outcome	$A\perp \perp Y_{1m}\mid C$	$✓$
	$A\perp \perp Y_0\mid C$		$✓$
Mediator-outcome	$M\perp \perp Y_{1m}\mid C,L,A=1$	$✓$
Positivity
Exposure condition	$\mathrm{P}\left(A=1\mid C\right)>0$	$✓$
	$\mathrm{P}\left(A=0\mid C,L\right)>0$	$✓$	$✓$
Mediator values	$\mathrm{P}\left(M=m\mid C,L,A=1\right)>0$	$✓$
Range of $m$ values	The support of distribution of $M$ given $C,L=l,A=0$ , where $l$ is the actual value of $L_1$	$✓$